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In single-variable calculus, we're introduced to the idea of best linear approximation, that is:

$$f(x) \approx f(a) + f'(a)(x - a)\tag{1}$$

And this is the best linear approximation for the point $a$. In multivariable calculus, the same concept is presented again, and for two variables, it is:

$$f\left(x,y\right)\approx f\left(a,b\right)+\frac{\partial f}{\partial x}\left(a,b\right)\left(x-a\right)+\frac{\partial f}{\partial y}\left(a,b\right)\left(y-b\right)$$

I've been thinking for a while but couldn't guess why this is important. We could say that in the $(1)$, it gives us the line tangent to the function $f$ at point $f(a)$, this would reveal us how much the function is increasing at that point - but the same can be said using only the derivative.

For multivariable calculus, it's even worse. They tell us that there is a best linear approximation, which is a plane tangent to a surface, but in this case, we are not told what they are important for: The subject is given and then we jump to other subjects.

For a while, I thought that for some functions, there is some points in which it is easy to calculate the value and other points in which it's hard to do it, ex: $\sqrt{4}=2,\sqrt{5}=2.23607\dots$ and then we could anchor to these points $a$, vary a little bit the $x$ and find an approximation, perhaps with an error function that allows an easier computation some way? But then I thought that this kind of problem could be easily bypassed with the use of modern computers. So after all, why are best linear approximations useful? I'm mostly interested in applications to mathematics, but whatever comes in mind, just say it.

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    $\begingroup$ The whole point of linear approximation is that linear maps are easy to understand (witness all the resources of the subject called linear algebra), but nonlinear ones aren't. One basic approach to understanding nonlinear dynamical systems, for example, is to linearize near equilibrium points. This is why chaos belongs to classical mechanics, not quantum. (Schrödinger's equation is linear. Newton's equations are generally nonlinear.) $\endgroup$ – symplectomorphic Aug 20 '16 at 4:03
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    $\begingroup$ Just in regards to computing square roots with modern computers, you might like to read up on the fast inverse square root approximation, which uses a combination of Newton iteration and linear approximations in gaming. $\endgroup$ – Mattos Aug 20 '16 at 4:05
  • $\begingroup$ Also, I think it was Raoul Bott who said: "Eighty percent of mathematics is linear algebra." $\endgroup$ – symplectomorphic Aug 20 '16 at 4:10
  • $\begingroup$ I wouldn't call that the best linear approximation. It's just a truncated Taylor series which gives a credible and defendable linear approximation near $a$. If you want the best linear approximation around $a$, you'd have to first define the neighborhood, and the criteria for best. For example, $x^2 \approx 2 x - 1$ would likely not be the best linear approximation for $f(x) = x^2$ on any interval $(1-\epsilon, 1+\epsilon)$ no matter how you define best. $\endgroup$ – dxiv Aug 20 '16 at 4:21

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